The average depth and heat flow of oceanic lithosphere as functions of age are well described by cooling plate models in which old lithosphere approaches an asymptotic thermal structure, causing average depth and heat flow to flatten. However, some areas are significantly shallower or deeper than the global average for their age. One possibility is that the deviations reflect variations in lithospheric temperature structure. Another is that the deviations reflect processes including excess volcanism or dynamic effects of mantle flow. The first hypothesis assumes that the average flattening reflects thermal perturbations to halfspace cooling, so the temperature structures of areas that are unusually deep for their age reflect continued halfspace cooling and thus should have lower heat flow. Although this hypothesis predicts lower heat flow at deeper sites in old lithosphere, the deep sites are divided approximately evenly between ones with high and low heat flow. Instead, the anomalously deep sites occur primarily at passive continental margins, perhaps because of dynamic topography due to sublithospheric mantle processes, and in only a few cases thinner crust formed at slow spreading rates immediately after rifting. Similarly, preferentially high heat flow is essentially not observed at anomalously shallow sites, primarily on hotspot swells, indicating that the swells do not result from hotspots significantly reheating the lithosphere. Thus, in general, neither shallow nor deep areas reflect primarily perturbed lithospheric thermal structure. Hence a plate model is more useful than a halfspace model in describing how ocean depth and heat flow vary with lithospheric age, and excluding the vast majority of the seafloor while ascribing significance to the small fraction matching the halfspace model is pointless.